ODE_Chapter 03-06 [Jan 2014]

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Ordinary Differential Equations [FDM 1023]

Transcript of ODE_Chapter 03-06 [Jan 2014]

Page 1: ODE_Chapter 03-06 [Jan 2014]

Ordinary Differential Equations[FDM 1023]

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Linear Higher-Order Differential Equations

Chapter 3

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Overview

Chapter 3: Linear Higher-Order Differential Equations

3.1. Definitions and Theorems

3.2. Reduction of Order

3.3. Homogeneous Linear Equations with

Constant Coefficients

3.4. Undetermined Coefficients

3.5. Variation of Parameters

3.6. Cauchy-Euler Equations

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3.6 Cauchy-Euler Equations

Learning Outcome

At the end of this section, you should be ableto:

Identify and solve a Cauchy-Eulerdifferential equation

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A linear Cauchy-Euler differential equation

���� ���

��� + �������� �����

����� +⋯+ ��� ���� + �� = �(�)

The coefficients �� , ���� , … ,� are constants

The degree of � is the same as the order of the derivatives

same same Function in terms of �

3.6 Cauchy-Euler Equations

same

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A linear second order Cauchy-Euler differential equation

same same Function in terms of �

3.6 Cauchy-Euler Equations

��� ���

��� + �� ���� + �� = �(�)

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Homogeneous Cauchy-Euler differential equation

3.6 Cauchy-Euler Equations

��� ���

��� + �� ���� + �� = 0

The general solution is

� = ��

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3.6 Cauchy-Euler Equations

��� ���

��� + �� ���� + �� = �(�)Non-Homogeneous Cauchy-Euler differential equation

The general solution is

� = �� + ��

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�� is obtained by solving the associated homogeneous

�� is obtained by using variation of parameters

��� ���

��� + �� ���� + �� = 0

��� ���

��� + �� ���� + �� = �(�)

3.6 Cauchy-Euler Equations

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������ + ���� + �� = 0

�� = �����

��� = � � − 1 ����

� = ��

3.6 Cauchy-Euler Equations

Method of Solution

Step 1: Let the solution of the DE be

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���� � − 1 ���� + ������� + ��� = 0

3.6 Cauchy-Euler Equations

Step 2: Substitute into DE

������ + ���� + �� = 0

�� � − 1 �� + ���� + ��� = 0��(� − 1) + �� + � �� = 0

��� + � − � � + � = 0

Auxiliary Equation

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3.6 Cauchy-Euler Equations

Step 2: Substitute into DE

������ + ���� + �� = 0

��� + � − � � + � = 0

DE

AE

Step 3: Solve the AE

Step 4: Find the general solution

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The general solution is

Case 1: Distinct and Real Roots

�� ≠ �

3.6 Cauchy-Euler Equations

� = ��!�� + � !�

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The general solution is

Case 2: Repeated Real Roots

�� = �

3.6 Cauchy-Euler Equations

� = ��!�� + � !�� "# !

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The general solution is

Case 3: Conjugate Complex Roots

�� = $ + %& , � = $ − %&

3.6 Cauchy-Euler Equations

� = !$ �� '() & "# ! + � )*# & "#!

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Solve �� +,-+., − 2� +-+. − 4� = 0

� = ���� = �����

��� = � � − 1 ����

Example 1

3.6 Cauchy-Euler Equations

Solution

Step 1: Let the solution of the DE be

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�� − 3� − 4 �� = 0

��� �− 1 ���� − 2������ − 4�� = 0� � − 1 �� − 2��� − 4�� = 0

3.6 Cauchy-Euler Equations

Step 2: Substitute into DE

�� ���

��� − 2� ���� − 4� = 0

�� − 3� − 4 = 0��� + � − � � + � = 0

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�� = −1 , �� = 4

= ����� + ���2

3.6 Cauchy-Euler Equations

Step 3: Solve the AE

�� − 3� − 4 = 0Case 1

Step 4: Find the general solution

The general solution is � = ����3 + ����,

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� = ���� = �����

��� = � � − 1 ����

Example 2

3.6 Cauchy-Euler Equations

Solve 4�� +,-+., + 8� +-+. + � = 0

Solution

Step 1: Let the solution of the DE be

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3.6 Cauchy-Euler Equations

Step 2: Substitute into DE

��� + � − � � + � = 0

4�� ���

��� + 8� ���� + � = 0

4�� + 4� + 1 = 0

4��� �− 1 ���� + 8������ + �� = 04� � − 1 �� + 8��� + �� = 0

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= ������ + ���

��� ln�

3.6 Cauchy-Euler Equations

Step 3: Solve the AE

Case 2

Step 4: Find the general solution

The general solution is � = ����3 + ����3 ln �

4�� + 4� + 1 = 02� + 1 � = 0

�� = �� = −12

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� = ���� = �����

��� = � � − 1 ����

Example 3

3.6 Cauchy-Euler Equations

Solve 4�� +,-+., + 17� = 0Solution

Step 1: Let the solution of the DE be

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3.6 Cauchy-Euler Equations

Step 2: Substitute into DE

��� + � − � � + � = 0

4�� − 4� + 17 = 0

4��� �− 1 ���� + 17�� = 04� � − 1 �� + 17�� = 0

4�� ���

��� + 17� = 0

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= ��� �� cos 2 ln � + �� sin 2 ln �

3.6 Cauchy-Euler Equations

Step 3: Solve the AE

Case 3

Step 4: Find the general solution

The general solution is � = �< �� cos = ln � + �� sin = ln �

�� = 12 + 2>, �� = 1

2 − 2>4�� − 4� + 17 = 0

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� = ���� = �����

��� = � � − 1 ����

Example 4

3.6 Cauchy-Euler Equations

Solve �@ +A-+.A + 5�� +,-+., + 7� +-+. + 8� = 0

���� = � � − 1 (� − 2)���@

Solution

Step 1: Let the solution of the DE be

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3.6 Cauchy-Euler Equations

Step 2: Substitute into DE

�@ �@�

��@ + 5�� ���

��� + 7� ���� + 8� = 0

�@� �− 1 � − 2 ���@ + 5��� �− 1 ����+7������ + 8�� = 0

�� � � − 1 � − 2 + 5� � − 1 + 7� + 8 = 0�� �@ + 2�� + 4� + 8 = 0

�@ + 2�� + 4� + 8 = 0

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3.6 Cauchy-Euler Equations

Step 3: Solve the AE

Case 3

Step 4: Find the general solution

The general solution is

�@ + 2�� + 4� + 8 = 0(� + 2)(�� + 4) = 0

�� = −2 , �� = 2> , �@ = −2>Case 1

� = ����3 + �< �� cos = ln � + �@ sin = ln �= ����� + �� cos 2 ln � + �@ sin 2 ln �

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Example 5

3.6 Cauchy-Euler Equations

Solve ����� − 3��� + 3� = 2�2D.SolutionStep 1: Find the complementary solution

Solve the associated homogeneous equation

Change to auxiliary equation.

����� − 3��� + 3� = 0

�� − 4� + 3 = 0

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= ��� + ���@

3.6 Cauchy-Euler Equations

The roots of the auxiliary equation are

�� − 4� + 3 = 0

� − 1 � − 3 = 0�� = 1 , �� = 3 Case 1

The complementary solution is

�� = ����3 + ����,

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3.6 Cauchy-Euler Equations

Step 2: Find the particular solution

Step 2.1: Identify �� and ���� = ��� + ���@�� = � , �� = �@

Step 2.2: DE in standard form and identify E ������ − 3��� + 3� = 2�2D.

��� − 3� �� +

3�� � = 2��D.

E(�) = 2��D.

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3.6 Cauchy-Euler Equations

F�� = G�G = −��E �

G F�� = G�G = ��E �

G

Step 2.3: Compute

and

G� =? G� =?G =?

G(�, �@) = � �@1 3��

= 2�@

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3.6 Cauchy-Euler Equations

G� = 0 �@2��D. 3��

= −2�ID.G� = � 0

1 2��D.= 2�@D.

G� = 0 ��E(�) ��� G� = �� 0

��� E(�)

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F�� = −2�ID.2�@

= −��D.

F�� = 2�@D.2�@

= D.

3.6 Cauchy-Euler Equations

F�� = G�G F�� = G�

G

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Step 2.4: Integrate

F�� = −��D. F�� = D.

F� = −J��D. ��

= −��D. + 2�D. − 2D.F� = JD. ��

= D.

3.6 Cauchy-Euler Equations

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Step 2.5: Particular solution

�� = F� � �� � + F�(�)��(�)

Step 3: General solution

The general solution is

� = �� + ��

= −��D. + 2�D. − 2D. � + D.�@= 2��D. − 2�D.

= ��� + ���@ + 2��D. − 2�D.

3.6 Cauchy-Euler Equations